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Theorem tlt2 29792
Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
tlt2.b 𝐵 = (Base‘𝐾)
tlt2.e = (le‘𝐾)
tlt2.l < = (lt‘𝐾)
Assertion
Ref Expression
tlt2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 < 𝑋))

Proof of Theorem tlt2
StepHypRef Expression
1 exmidd 431 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ∨ ¬ 𝑋 𝑌))
2 tlt2.b . . . . 5 𝐵 = (Base‘𝐾)
3 tlt2.e . . . . 5 = (le‘𝐾)
4 tlt2.l . . . . 5 < = (lt‘𝐾)
52, 3, 4tltnle 29790 . . . 4 ((𝐾 ∈ Toset ∧ 𝑌𝐵𝑋𝐵) → (𝑌 < 𝑋 ↔ ¬ 𝑋 𝑌))
653com23 1291 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 < 𝑋 ↔ ¬ 𝑋 𝑌))
76orbi2d 738 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 < 𝑋) ↔ (𝑋 𝑌 ∨ ¬ 𝑋 𝑌)))
81, 7mpbird 247 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995  ltcplt 16988  Tosetctos 17080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-preset 16975  df-poset 16993  df-plt 17005  df-toset 17081
This theorem is referenced by:  tlt3  29793  archirngz  29871  archiabllem2a  29876
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